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Chapter 9: Problem 47

Show that if \(C_{0}+C_{1} x+C_{2} x^{2}+C_{3} x^{3}+\cdots\) converges for \(|x|

Short Answer

Expert verified
The derived series converges for \(|x|<R\) because the factor \((n+1)/n\) does not affect the limit \(L|x|<1\).

Step by step solution

01

Apply the Ratio Test to the Original Series

The original series is \( \sum_{n=0}^{\infty} C_n x^n \). By the ratio test, this series converges if \( \lim_{n \to \infty} \left| \frac{C_{n+1} x^{n+1}}{C_n x^n} \right| < 1 \). Simplifying, we get: \( \lim_{n \to \infty} \left| \frac{C_{n+1}}{C_n} \right| |x| < 1 \). Let \( L = \lim_{n \to \infty} \left| \frac{C_{n+1}}{C_n} \right| \), then the series converges for \( |x| < \frac{1}{L} = R \).
02

Consider the Derived Series and Apply the Ratio Test

The derived series is \( \sum_{n=1}^{\infty} n C_n x^{n-1} \). For convergence, consider its terms as \( a_n = n C_n x^{n-1} \). The ratio of successive terms is \( \frac{a_{n+1}}{a_n} = \frac{(n+1)C_{n+1}x^n}{nC_n x^{n-1}} = \left(\frac{n+1}{n}\right) \left(\frac{C_{n+1}}{C_n}\right)x \).
03

Analyze the Limit of the Ratio

The crucial part is to examine the limit \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty} \left| \frac{n+1}{n} \right| \left| \frac{C_{n+1}}{C_n} \right| |x| = \lim_{n \to \infty} \left( 1 + \frac{1}{n} \right) L |x| \). Since \( \left( 1 + \frac{1}{n} \right) \to 1 \) as \( n \to \infty \), we have \( L |x| < 1 \).
04

Conclude the Convergence of the Derived Series

Since \( L |x| < 1 \) as \(|x| < R \), it follows that the derived series \(C_{1}+2C_{2}x+3C_{3}x^{2}+\cdots\) converges for \(|x|<R\). The factor of \(n+1/n\) does not affect convergence as \(n\to \infty\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Convergence of Series
When we talk about the convergence of a series, we're exploring whether an infinite sum of numbers results in a finite value. Imagine adding numbers infinitely and still getting to a specific number! The Ratio Test comes into play here, providing a systematic way to check for this.
  • The Ratio Test looks at the absolute values of the terms in the series.
  • It compares the size of consecutive terms as the index goes to infinity. \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \).
  • If this limit is less than 1, the series converges absolutely.
Using this test helps avoid the difficulty of adding up infinitely many terms directly, which is counterintuitive. It does so by focusing on the behavior of term ratios which gradually reveal the behavior of the entire series.
In the original exercise, this aspect was crucial to show that the adjusted series maintained its convergence for \(|x| < R\).
Power Series
A power series is like a polynomial, but with infinitely many terms. Each term involves a variable, usually denoted as \(x\), raised to successively higher powers and multiplied by coefficients.
  • It generally looks like: \( C_0 + C_1 x + C_2 x^2 + C_3 x^3 + \cdots \).
  • The radius of convergence \( R \) is crucial because it defines where this series behaves well.
  • In a proper interval, \(|x| < R\), a power series behaves like a regular function鈥攏otably with convergence properties.
In the exercise solution, knowing the original series conferred convergence over a radius \( R \) immediately suggested that the modified series might have similar properties. This arises from the consistent patterns in power series, where alterations to coefficients align with \( x \)'s powers but still respect this territorial convergence restriction.
Limit of Sequence
The limit of a sequence is about understanding what happens to the terms as they progress towards infinity. Does the sequence settle into a specific value, or does it bounce around endlessly? When using the limit of a sequence within the scope of a series convergence discussion, it's important to understand:
  • The limit provides a snapshot of long-term behavior.
  • In terms of the Ratio Test, we often examine \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \).
  • If this tends towards a value less than 1, the series converges.
In our exercise, the checker \( \lim_{n \to \infty} \left( 1 + \frac{1}{n} \right) L |x| \) gives great insight. As \(n\) grows, this shifts towards \(L|x|\), reinforcing the series' behavior given the pre-defined bounds, \(|x| < R\). Functionally, this means although terms grow and change, the limit parameter conditions the series to a secure pattern.

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Most popular questions from this chapter

True or false. Give an explanation for your answer. \(-5 < x \leq 7\) is a possible interval of convergence of a power series.

Determine whether the series converges. $$\sum_{n=1}^{\infty} \frac{5 n+2}{2 n^{2}+3 n+7}$$

The series converges. Is the sum affected by rearranging the terms of the series? $$\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n}$$

Explain what is wrong with the statement. The series \(\sum_{n=1}^{\infty} 1 /\left(n^{2}+1\right)\) converges by the ratio test.

The series converge by the alternating series test. Use Theorem 9.9 to find how many terms give a partial sum, \(S_{n},\) within 0.01 of the sum, \(S,\) of the series. $$\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n}$$
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